Restricted maximum likelihood estimation of genetic parameters accounting for genomic relationships has been reported to impose computational burdens which typically are many times higher than those of corresponding analyses considering pedigree based relationships only. This can be attributed to the dense nature of genomic relationship matrices and their inverses. We outline a reparameterisation of the multivariate linear mixed model to principal components and its effects on the sparsity pattern of the pertaining coefficient matrix in the mixed model equations. Using two data sets we demonstrate that this can dramatically reduce the computing time per iterate of the widely used 'average information' algorithm for restricted maximum likelihood. This is primarily due to the fact that on the principal component scale, the first derivatives of the coefficient matrix with respect to the parameters modelling genetic covariances between traits are independent of the relationship matrix between individuals, i.e. are not afflicted by a multitude of genomic relationships.